Spin Meissner effect in superconductors and the origin of the Meissner effect

J.E. Hirsch, UCSD Hvar, 2008

Why the Meissner effect is not understood, and how it can be

understood

Spin Meissner effect: spontaneous spin current in the ground

state of superconductors

Charge expulsion, charge inhomogeneity in superconducting state

Electrodynamic (London-like) equations for charge and spin

Experiments

3 key pieces of the physics that BCS theory got right:* Cooper pairs

* Macroscopic quantum coherence* Electron-phonon-induced attraction between electrons

* Energy gap

3 key pieces of the physics that BCS theory got right:* Cooper pairs

* Macroscopic quantum coherence* Electron-phonon-induced attraction between electrons

* Energy gap

(1) Key role of electron-hole asymmetry

(2) Key role of kinetic energy lowering as driving force

(4) Key role of spin-orbit interaction

(5) Key role of mesoscopic orbits

1988-2008

http://physics.ucsd.edu/~jorge/hole.html

(3) Macroscopic charge inhomogeneity and internal E-field

(6) Spontaneous currents in the absence of applied fields

Meissner effect: expulsion of magnetic field from interior of superconductor

1933The expulsion of magnetic flux from the interior of a superconducting metal when it is cooled in a magnetic field to below the critical temperature, near absolute zero, at which the transition to superconductivity takes place. It was discovered by Walther Meissner in 1933, when he measured the magnetic field surrounding two adjacent long cylindrical single crystals of tin and observed that at ?452.97°F (3.72 K) the Earth's magnetic field was expelled from their interior. This indicated that at the onset of superconductivity they became perfect diamagnets. This discovery showed that the transition to superconductivity is reversible, and that the laws of thermodynamics apply to it. The Meissner effect forms one of the cornerstones in the understanding of superconductivity, and its led F. London and H. London to develop their phenomenological electrodynamics of superconductivity.The magnetic field is actually not completely expelled, but penetrates a very thin surface layer where currents flow, screening the interior from the magnetic field.

cool

L=London penetration depth

same final state

EFaraday

I I

super normal

cool

apply B

two pathways to Meissner current

expel B

Faraday electric field points in opposite directions

Meissner current I points in the same direction

cool

cool

super normal

apply B

B

I

Why the Meissner effect is a puzzle

What is the 'force'pushing the electrons near the surface tostart moving all in the same direction, opposite to eEFarad ?

How is angular momentum conserved???

Meissnerstate

Current develops'spontaneously' uponcooling or lowering Hopposing EFarad

I

Lower the temperature...or lower slightly the applied H...

B=0

EFarad

B

The key to the Meissner effect

L

I

vs

R

Angular momentum in Meissner current:

(h=cylinder height, ns=superfluid density)

€

Le = (2πRλ Lhns) × (mevsR)

€

=(πR2hns) × (mevs(2λ L ))

r=2L orbits!

B

€

mevsR =angular momentum of 1 el

€

2πRλ Lhns =# of electrons in surface layer

2L

€

mevs(2λ L )

€

πR2hns

bulk

some very complicated math. . . ..

cylinder

Some simple relations:

€

=−1

3(

eh

2mec)2(

3nme

h2kF2

)

Density of states at the Fermi energy:

€

g(εF ) =3n

2εF

=3nme

h2kF2

Normal state: Superconducting state:London penetration depth:

€

1

λ L2

=4πnse

2

mec2

€

χLandau = −1

3μB

2 g(εF )

Magnetic susceptibility: Magnetic susceptibility:

€

χLondon = −1

4π

€

=−ne2

4mec2

< (kF−1)2 >

€

=−nse

2

4mec2

< (2λ L )2 >

€

=−1

4

nse2

mec2

4mec

2

4πnse2

Larmor diamagnetism

rv

Apply magnetic field B:

€

rE ⋅d

r l = −

1

c∫ ∂

∂t

r B ∫ ⋅d

r a

€

=>2πrE = −1

c

∂B

∂tπr2

magnetic susceptibility per unit volume: n=electrons/unit vol

€

χLarmor = nΔμ

B= −

ner2

4mec2

€

=−ne2

4mec2

< (kF−1)2 >

€

=−nse

2

4mec2

< (2λ L )2 >or

B

v

€

=>me

dv

dt= eE = −

er

2c

∂B

∂t

€

=>v = −er

2mecB

Orbital magnetic moment:

€

μ =e

2mecl =

e

2mecmevr =

er

2cv

€

=>μ =er

2cΔv = −

er

2c

er

2mecB = −

e2r2

4mec2

B

€

χLarmor = nΔμ

B= −

ne2

4mec2

r2

€

=−nse

2

4mec2

< (2λ L )2 >

How the transition occurs

2Lorbit expansion: kF-1

r=2L orbits

€

χLarmor = −nse

2

4mec2

< r2 >B

r

Normal state: Superconducting state:

€

χLandau = −1

3μB

2 g(εF )

€

=−nse

2

4mec2

< (kF−1)2 >

€

χLondon = −1

4π

r=kF-1 orbits

same final state

EFaraday

I I

super normal

cool

apply B

two pathways to Meissner current

expel B

€

A = λ LB

€

< p >= 0

The two pathways to the Meissner current

rv

Apply magnetic field B:

€

rE ⋅d

r l = −

1

c∫ ∂

∂t

r B ∫ ⋅d

r a

Expand electron orbit in B:

Faraday's law pushes e- Lorentz force pushes e-

B

v

€

=>v = −er

2mecB

€

=>me

dv

dt= eE = −

er

2c

∂B

∂t

€

=>vφ = −er

2mecB

r=2L

€

rF =

e

c

r v ×

r B +

r F r = me

dr v

dt

€

d

dt(r r ×

r v ) = −

e

2mec(r r ⋅

r v )

r B = −

e

2mec

d

dt(r2)

r B

vEv

B

F r

€

v =1

m(p −

e

cA)

BCS:

€

=>v = vφ = −eλ L

mecB

r=2L orbitsr=kF-1 orbits

Why is there macroscopic phase coherence in superconductors?

Normal stateNon-overlapping orbitsRelative phase doesn't matter

Superconducting stateHighly overlapping orbitsPhase coherence necessaryto avoid collisions

==>

€

:L2 =

mec2

4πe2×

h2

mee2

× a02 =

1

4π

hc

e2a0

⎡ ⎣ ⎢

⎤ ⎦ ⎥

2

€

:L2 =

mec2

4πnse2

, ns =1

a03

€

a0 =h2

mee2,

137

A little help from a friend...

==>

€

:L2 =

mec2

4πe2×

h2

mee2

× a02 =

1

4π

hc

e2a0

⎡ ⎣ ⎢

⎤ ⎦ ⎥

2

€

:L2 =

mec2

4πnse2

, ns =1

a03

€

a0 =h2

mee2,

137

A little help from a friend...

The speed of light must enter into the superconducting wave function!

So we learn from the Meissner effect that:transition to superconductivity = expansion of electronic orbit from r=kF

-1 to r=2L

What happens when there is no magnetic field?

Spin-orbit force deflects electron in expanding orbit!

Spin orbit scattering (Goldberger&Watson)

scatteringcenter scattering

center

μv p

€

rp =

r v

c×

r μ a moving magnetic moment is

equivalent to an electric dipole

spin-orbit spin-orbit

μ μ

vμ

So we learn from the Meissner effect that:transition to superconductivity = expansion of electronic orbit from r=kF

-1 to r=2L

What happens when there is no magnetic field?

Spin-orbit force deflects electron in expanding orbit!μ

v p

€

rp =

r v

c×

r μ

€

dr L

dt=

r τ =

r p ×

r E

E

€

me

d

dt(r r ×

r v ) = (

r v

c×

r μ ) ×

r E =

1

c(

r E ⋅

r v )

r μ

€

with r E = α

r r : me

d

dt(r r ×

r v ) =

α

c(r r ⋅

r v )

r μ =

α

2c

d

dt(r2)

r μ

€

=>vφ =E

2mecμB

€

=> r

r ×r v =

α

2c(r2)

r μ =

Er

2c

r μ

μ

μ

€

vφ

€

vφ

€

=>vφ =E

2mecμB

€

E = 2πρr, ρ =| e | ns

€

=>vφ =2πensr

2mecμB =

πensr

mec

eh

2mec

€

; with 4πnse

2

mec2

=1

λ L2

€

vφ =h

8me

r

λ L2

€

vφ =h

4meλ L

For r=2L

€

==> L = mevφr = . . ..

€

=h2

!!!!!!!!

Eμ

μ

€

vφ

€

vφ

What's E?

The two pathways to the Spin Meissner current

rv

'Apply' electric field E: Expand electron orbit in E:Maxwell's law pushes μ Lorentz torque pushes μ

μ

v

r=2L

v

E(t)

r

€

r =d

r L

dt= me

d

dt(r r ×

r v ) =

e

c

r r × (

r v ×

r B eff )

€

r =

rp ×

r E = (

r v

c×

r μ ) × (2π | e | ns

r r )

€

rB eff = 2πns

r μ

€

=>vφ = −er

2mecBeff = −

πnseμB

mecr

€

==> L = mevφr =h

2

€

=>me

dr v

dt=

r ∇(

r μ ⋅

r B ) =

1

2

r μ × (

r ∇ ×

r B )

€

=1

2c

∂

∂t(r μ ×

r E )

€

=>v = −1

2mec|r μ ×

r E |= −

πnseμB

mecr

Bμ

v

€

=>v = vφ = −2πnseμB

mecλ L = −

hπnse2

me2c 2

λ L = −h

4meλ L

€

r∇ ×

rB =

1

c

∂r E

∂t

Ground state of a superconductor

r=2L orbits r=2L orbits

spin down electrons spin up electrons

Currents in the interior cancel out, near the surface survive

==> there is a spontaneous spin current in the ground state of superconductors!

There is a spontaneous spin current in the ground state ofsuperconductors, flowing within L of the surface

€

rμ = eh

2mec

r σ

For L=400A, vcm/s

# of carriers in the spin current: ns

When a magnetic field is applied:

μ

μnv0

€

rv σ 0 = −

h

4meλ L

r σ × ˆ n

€

rv σ =

r v σ 0 −

e

mecλ L

r B × ˆ n

The slowed-down spin component stops when

€

B =mec

eλ L

vσ 0

€

=hc

4eλ L2

€

=φ0

4πλ L2

~ Hc1!

€

φ0 = hc / 2 | e |

v

B

no external fields applied

(JEH, EPL81, 67003 (2008))

Summary of argument:

1)

€

1

λ L2

=4πnse

2

mec2 (Ampere, Faraday, Newton, London)

3) Magnetic moment of electron is

€

μB =eh

2mec

Therefore: Superconductivity is an intrinsically relativistic effect

Electron spin and associated magnetic moment plays a key role The wavefunction of a superconductor contains c=speed of light

1)+2)+3)+4) ==>

€

L = h /2

1)+2)+3)+4) ==> magnetic field that stops the spin current is Hc1

2) Orbits have radius (to explain origin of Meissner current)

€

2λ L

4) Background positive charge density is = - superfluid dens.

€

| e | ns

What makes electrons move in the direction needed to create all these currents when T is lowered from above to below Tc?

B

Intermediatestate

B

I

Meissnerstate

I

B

I

Vortex state

Back to: cooling a superconductor in the presence of a B-field:A clue from plasma physics

www.mpia-hd.mpg.de/homes/fendt/Lehre/Lecture_OUT/lect_jets4.pdf

A clue from plasma physics

B

I

Meissnerstate

Intermediatestate

LeB

Vortex state

B

ILe

ve

€

rF B =

e

c

r v ×

r B

v

FB

v FB v

FB

Electrons have to flow away fromthe interior of the superconductor,towards the surface and towardsthe normal regions!

But if there is charge flow, it will result in charge inhomogeneity andan electric field in the interior of superconductors.

€

r∇ ⋅

rE = 4πρ

But if there is charge flow, it will result in charge inhomogeneity andan electric field in the interior of superconductors.

Can there be an electric field inside superconductors?

€

∂J

∂t=

ne2

mE free acceleration of electrons

London says NO. First London equation (1934):

€

J = nev (n=density, v=speed, J=current)

If E = 0, J increases to infinity, unless Newton’s law is violated?

€

mdv

dt= eE

€

r∇ ⋅

rE = 4πρ

€

dr v

dt=

∂r v

∂t+

r ∇v 2

2−

r v × (∇ ×

r v ) !

€

ne∂v

∂t=

ne2

mE + ne(

r v × (∇ ×

r v ) −

r ∇v 2

2)

€

∂J

∂t=

€

=ne2

m(E +∇φ)

€

∂J

∂t=

€

∂J

∂tcan be zero even if E is non-zero!

€

∂J

∂t=

ne2

mE

New electrodynamic equations for superconductors (JEH, PRB69, 214515 (2004))

1)

2)

€

∇⋅A +1

c

∂φ

∂t= 0 ; (Lorenz gauge)

€

(r, t) − ρ 0 = −1

4πλ L2

[φ(r, t) − φ0(r)]

€

φ0(r) = d3∫ r'ρ 0

| r − r' |==>

€

E = −∇φ −1

c

∂A

∂t Note: ==>

€

J = −ne2

mcA = −

c

4πλ L2

A ; 1

λ L2

≡4πne2

mc 2

€

∂J

∂t= −

ne2

mc

∂A

∂t=

ne2

m(E +∇φ) , NOT

€

∂∂t

= −1

4πλ L2

∂φ

∂t

, continuity equation:

€

∇⋅J +∂ρ

∂t= 0 ==>

€

∇⋅J = −c

4πλ L2∇ ⋅A

integrate in time, 1 integration constant 0 , ...

Electrodynamics

€

∇2B =1

λ L2

B +1

c 2

∂ 2B

∂t 2

€

∇2J =1

λ L2

J +1

c 2

∂ 2J

∂t 2

€

∇2(E − E0) =1

λ L2

(E − E0) +1

c 2

∂ 2(E − E0)

∂t 2

€

∇2(ρ − ρ 0) =1

λ L2

(ρ − ρ 0) +1

c 2

∂ 2(ρ − ρ 0)

∂t 2

Relativistic form: 2

€

≡∇2 −1

c 2

∂ 2

∂t 2

or equivalently

€

J − J0 = −c

4πλ L2

(A − A0)€

(A − A0) =1

λ L2

(A − A0)2

€

A = (r A (

r r , t),iφ(

r r , t))

€

J = (r J (

r r , t),icρ(

r r , t))

€

J0 = (0,icρ 0)

€

A0 = (0,iφ0(r r ))

€

r∇ ⋅

rE 0 = 4πρ 0

;

€

∇2φ(r) = 0 outside supercond.

+assume (r) and its normal derivative are continuous at surface

Electrostatics:

€

∇2φ(r) = −4πρ(r)

€

∇2φ0(r) = −4πρ 0

€

∇2(φ(r) − φ0(r)) =1

λ L2

(φ(r) − φ0(r))

€

∇2(ρ(r) − ρ 0) =1

λ L2

(ρ(r) − ρ 0)

€

∇2(r E −

r E 0) =

1

λ L2

(r E −

r E 0)

L

€

(r) = ρ 0(1−R3

3λ L2

sinh(r /λ L )

R /λ L cosh(R /λ L ) − sinh(R /λ L ))

Solution for sphere of radius R:

No electric field outside sphere

Sample size dependence of expelled charge (Q) and E-field

< 0 = charge density near surface

> 0 = charge density in interior

Q ~ R3 ~ - R2 L

Electrostatic energy cost:

UE ~ Q2/R ~ ( R2 L)2/R ~ ( R3 ~ ( R5 ~ Volume~R3

==> independent of R, ~ 1/R

sphere of radius R

Electric field vs. r: E

Em

r R1 R2

€

Em = −4πλ Lρ−

independent of R

how big is - , Em

?

L

How much charge is expelled?

element Tc(K) Hc(G) L(A) Extra elec- trons/ion

Em

(Volts/cm)

Al 1.14 105 500 1/17 mill 31,500

Sn 3.72 309 510 1/3.7 mill 92,700

Hg 4.15 412 410 1/2.5 mill 123,600

Pb 7.19 803 390 1/1 mill 240,900

Nb 9.50 1980 400 1/1.3 mill 308,400

Emax

RR

Spin currents in superconductors (JEH, Phys. Rev. B 71, 184521 (2005))

€

ck↑+ c−k↓

+carries a spin current

€

< ck↑+ c−k↓

+ >≠< c−k↑+ ck↓

+ > necessarily in the presence of internal E-field

Internal electric field (in the absence of applied B) pointing out

€

Jch arg e =n

2(v↑ + v↓ ) = 0 no charge current ==> no B-field

€

Jspin =n

2(v↑ − v↓ ) ≠ 0 spin current without charge current!

E

Flows within a London penetration depthof the surface

Speed of spin current carriers:~ 100,000 cm/s

Number of spin current carriers:=superfluid density

€

0 = −2λ L

Rρ−

€

vσ 0 =h

4meλ L

σ

How much charge is expelled?

We now have 2 new pieces of physics of superconductors:

r=2L orbits

How are they related?

L0

€

Em = −4πλ Lρ−spin current

charge expulsion

€

L = h /2

€

vσ 0 =h

4meλ L

σ ==>

~ Hc1

(JEH)

€

Em = −hc

4eλ L2

=φ0

4πλ L2

~ Hc1

€

−=nsevσ 0

c

€

ns(1

2mevσ 0

2 ) =Em

2

8π

€

Jσ (r r ) ≡ ensvσ (

r r ) = −

c

8πλ L

r σ × (

r E (

r r ) −

r E 0(

r r ))

€

r∇ ×Jσ (

r r ) = −

c

2λ L

(ρ(r r ) − ρ 0) μ

μnv0

€

vσ 0 =h

4meλ L

σ

(Recall

€

ns(1

2mevs

2) =B2

8π)

€

−

€

€

Em = −4πλ Lρ− (charge neutra-lity)

€

=hc

16πeλ L3

Emax

R

Spin current electrodynamics (4d formulation)

Energetics

€

vσ 0 =h

4meλ L

σ

€

εpair = 2 ×1

2me (vσ

0 )2

Apply a magnetic field:

€

v↑ = v↓ = vσ0

€

v↑ → 2vσ0 , v↓ → 0, ε pair →

1

2me (2vσ

0 )2 = 4 ×1

2me (vσ

0 )

==> condensation energy per particle:

€

εc =1

2me (vσ

0 )2

energy lowering per particle in entering sc state:

€

2εc2εc = εc + εc

Coulomb energy cost + condensation energy

€

ns(1

2mevσ 0

2 ) =Em

2

8π

€

1

2mevσ 0

2 =1

4

h2

2me(2λ L)2

€

=π2

nsμ B2

€

~ 1.5μeV~ condensation energy of sc

Type I vs type II materials=distance between orbit centers

Type I: > 2L

Type II: < 2L

Phase difference:

€

hπ =e

cΦB ==> ΦB =

hc

2e≡ Φ0

€

Φ0 =hc

(2e)

€

=h

2

⎛

⎝ ⎜

⎞

⎠ ⎟c

e

€

rp = me

r v +

e

c

r A ,

r p =

h

i

r ∇

€

hϑ =me

r v ⋅d

r l ∫ +

e

cΦB

What drives superconductivity?

1) Excess negative charge (CuO2)=, (MgB2)-, (FeAs)-

3) Kinetic energy lowering

2) Almost full bands (hole conduction in normal state)

(Kinetic energy is highest when band is almost full)

kF-1 is small

kF

too many electrons!

How is angular momentum conserved in the Meissner effect??

Electromagnetic field carries angular momentum!

=-Le

But - is way too small to give enough Lfield

Spin-orbit interaction transfers counter-L to ions!

JEH, J. Phys.: Condens. Matter 20 (2008) 235233

B

B

Le

Lfield

Lions

Experimental tests?

1) Detect spin current μ

μnv0* polarized light scattering (PRL100, 086603 (08)

* inelastic polarized neutron scattering

* photoemission

* Detect electric fields produced by spin current

* Insert a 'spin current rectifier'

2) Detect internal electric field

3) Response of superconductor to applied electric field

4) Detect change in plasmon dispersion relation in sc state

.....

* positrons, muons, neutrons

(Tao effect)

To prove this theory wrong, find clear experimentalevidence for any of the following:

* A superconductor that has no hole carriers in normal state

* A superconductor that has no outward-pointing electric field in its interior

* A superconductor that has no spontaneous spin current near the surface, with carrier density ns/2 and speed

€

v = h /4meλ L

* A superconductor with tunneling asymmetry of intrinsic origin that has opposite sign to the one usually observed

* A superconductor with gap function that has no εk dependence

* A superconductor that expels magnetic fields without expelling negative charge

* A high Tc superconductor with no excess negative charge anywhere

* A superconductor not driven by kinetic energy lowering

IT IS A FALSIFIABLE THEORY!